On a testing-function space for distributions associated with the Kontrovich-Lebedev transform

Semyon B. Yakubovich

Resum

We construct a testing-function space, which is equipped with the topology that is generated by $L_{v,p}$-multinorm of the differential operator $A_x = x^2-{x {d\over dx} {[x {d\over dx}]}}$, and its $k$-th iterates $A^k_x$, where $k$ = 0, 1, . . . , and $A^0_x\varphi = \varphi$. Comparing with other testing-function spaces, we introduce in its dual the Kontorovich-Lebedev transformation for distributions with respect to a complex index. The existence, uniqueness, imbedding and inversion properties are investigated. As an application we find a solution of the Dirichlet problem for a wedge for the harmonic type equation in terms of the Kontorovich-Lebedev integral.